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  1. 1. November 30, 2001 MEASURING FINANCIAL CASH FLOW AND TERM STRUCTURE DYNAMICS AbstractFinancial turbulence is a phenomenon occurring in anti - persistent markets. In contrast, financial crisesoccur in persistent markets. A relationship can be established between these two extreme phenomena oflong term market dependence and the older financial concept of financial (il-)liquidity. The measurementof the degree of market persistence and the measurement of the degree of market liquidity are related.To accomplish the two research objectives of measurement and simulation of di¤erent degrees of financialliquidity, I propose to boldly reformulate and reinterpret the classical laws of fluid mechanics into cashflow mechanics. At first this approach may appear contrived and artificial, but the end results of thesereformulations and reinterpretations are useful quantifiable financial quantities, which will assist us withthe measurement, analysis and proper characterization of modern dynamic financial markets in ways thatclassical comparative static financial - economic analyses do not allow.Cornelis A. Los, Ph.D.Kent State University, College of Business Administration BSA416, & Graduate School of Management, Kent, OH 44242, Utel: 1-330-672-1207; fax: 1-330-672-9806; e-mail: clos@bsa3.kent.eduKey words: cash flow dynamics; term structure dynamics; financial intermittency; Reynolds number; multiresolution analys
  2. 2. 1 IntroductionPeters (1989, 1994) of PanAgora Management suggests that, to understand financial turbulence,the dynamics of cash flows between the various market participants, within and between di¤erentasset markets, should be analyzed and measured more carefully. Although there exists not yet acomplete theory of physical turbulence, let alone a theory of financial turbulence, many parallelsbetween the two phenomena have been noted by, for example, Mandelbrot (1982, 1998). Simultaneously, the accurate measurement of financial illiquidity and of financial illiquidity riskhas gained in importance, as the example of the tax - payer financed bail - out of the collapsedLong Term Capital Management (LTCM) hedge fund in 1998 demonstrates.1 This hedge fundapplied a trading strategy known as convergence arbitrage, which is based on the idea that if twosecurities have the same theoretical price, because they have the same return - risk profile, theirmarket prices should eventually be the same. But this convergence strategy ignores the observationthat financial risk is a time - dependent phenomenon and not a time - independent phenomenonto which the usual central limit theory based on i.i.d. assumptions applies. Indeed, in the summerof 1998 LTCM made a huge $4 billion loss. This was triggered when Russia defaulted on its debt,which caused a flight to quality in the German bond market. LTCM itself did not have a large exposure to Russian debt, but it tended to be long illiquidGerman (o¤ - the - run) bonds and short the corresponding liquid German (on - the - run) bonds.The spreads between the prices of the illiquid bonds and the corresponding liquid bonds widenedsharply after the Russian default. Credit spreads also increased and LTCM was highly leveraged,with a debt/equity ratio of close to 300, so that the loss of profits on its hedges immediatelytranslated in a very rapid vanishing of its market valued equity and threatened bankruptcy. Whenit was unable to make its projected ”risk - free” arbitrage profits, it experienced huge losses andthere were margin calls on its positions that it was unable to meet. 1 This example is borrowed from Hull, 2001, p. 428 - 429. Cf. also Dunbar (2000) and Jorion (1999). 1
  3. 3. But LTCM did not operate in a vacuum. It operated in a particular financial environment.LTCM’s position was made more di¢cult by the fact that many other hedge funds followed similarconvergence arbitrage strategies. When LTCM tried to liquidate part of its portfolio to meet itsmargin calls, by selling its illiquid o¤ - the - run bonds and by buying its liquid on - the runbonds, other hedge funds faced with similar problems tried to do similar trades. The price ofthe on - the - run bonds rose relative to the price of the o¤ - the -run bonds. This caused theilliquidity spreads to widen even further and to reinforce the flight to quality. Thus the illiquidityproblem was exacerbated and not alleviated. This was a clear example of long - term illiquidity riskdependence that was not foreseen by the time independence assumptions of geometric Brownianmotion arbitrage models used by LTCM’s managers and partners.2 Despite its obvious importance, the proper empirical measurement and analysis of liquidityand of illiquidity risk is still in its infancy. For example, there is still no agreement on howfinancial market liquidity should be measured. More precisely stated: there does not yet exista measurement standard for the various degrees of financial liquidity. Which levels of financialilliquidity are prone to generate financial catastrophes, which levels of financial illiquidity allowregular, liquid trading activity, and which levels of ultra financial liquidity are prone to generatefinancial turbulence? Therefore, in this paper we discuss the empirical measurement of the dynamics of marketilliquidity, in particular of the illiquidity of cash flows, which is directly related to the dynamics ofthe term structure of rates of return on cash investments and to the concept of (bond and equity)duration. For a recent survey of current theoretical work on term structure dynamics, see the recentarticle by Hu (2001). For a thorough discussion of a few current approaches to the empirical 2 Some of these partners which were the same Nobel Prize winners, who had proposed this theory in 1973.At least, they had courageously put their money where there mouthes were. But the empirical experiment of1998 drastically refuted the non-empirical assumptions of their theory......”A Trillion Dollar Bet.” is an excellentpedagogic video documentary of the rise and fall of LTCM, of the crucial role the Nobel prize winning Black-Scholessolution model in this financial market a¤air, and of the regulatory aftermath of LTCM’s bail-out by the Fed. 2
  4. 4. model identification of term structure dynamics, in particular of its principal component analysis,see the companion article by Chapman and Pearson (2001). In 1989 we thoroughly criticised andrejected principal component analysis, since it was non - scientifically subjective (Los, 1989). To accomplish the two research objectives of measurement and simulation of various degreesof financial liquidity illiquidity, I propose to boldly reformulate and reinterpret the classical lawsof fluid mechanics into financial cash flow mechanics, in particular, the laws of conservation ofinvestment capital (= ”mass”), of the financial cash flow rate (= ”momentum”) and of therate of financial cash return (= 2£ ”energy”). My newly proposed financial cash flow laws areunashamedly modeled on the basic conservation laws of hydraulics and aerodynamics.3 For example, we’ll study a ”scale - by - scale cash flow rate of return budget equation,” whichallows us to interpret the transfer of cash returns among di¤erent scales of investment cash flow(cf. Los, 1999). For more detailed background on the formulation of these laws in the mechanicsof incompressible fluids, see Batchelor (1970), Landau and Lifshitz (1987) and Tritton (1988). VanDyke ’s (1982) album provided me with inspiring visualizations of physical fluid dynamics andturbulence. At first this approach may appear contrived and artificial, but the end results of these re-formulations and reinterpretations are various useful quantifiable financial quantities, which willassist us with the measurement, analysis and characterization of the modern dynamic financial 3 A flow dynamics approach to the cash flows in financial markets is not new in economics. Compare, for example,the physical income flow dynamics modeling of an economy in the 1950s at the London School of Economics andPolitical Science (LSE) by the engineer and economist A. W. Phillips, familiar from the (expectations - augmented)Phillip’s Curve in undergraduate textbooks on macro-economics. Unfortunately, the financial economic researchof markets has focused on the static ”equilibrium theory ” of markets of Nobel Memorial prize winners GerardDebreu and Kenneth Arrow, while an dynamic ”transport theory ” has mostly been neglected. Consequently, thedynamics of cash flows has not been further developed since Phillips’ heuristic macro - economics research. Butfluid dynamics has a long mathematical history going back to Dutch mathematician and engineer Simon Stevin’s Hydrostatics (1605), Italian Evangelista Torricelli’s law of e- ux (1644), and French Blaise Pascal’s uniformpressure law of hydrostatics (1653). Sir Isaac Newton devoted over one quarter of his famous Principia book tothe analysis of fluids (Book II, 1713) and added some original ideas, like his hypothesis of viscosity. This wasfollowed by the mathematical explanations of the mechanics of ideal, frictionless fluid mechanics by Daniel andJacques Bernouilli (1738) and Leonhard Euler (1755). In 1827, Claude Navier derived the equations of viscous flow,which were published by Sir George Gabriel Stokes in 1845, the celebrated Navier - Stokes equations. Work byJoseph Boussinesq (1877) and Osborne Reynolds (1883) on laminar (streamlined) flow and turbulent (erratic) flowextended these Navier - Stokes equations to turbulent flow, by including the Reynolds stresses and measurements,to be discussed in this paper. 3
  5. 5. markets. They will do that in ways that could not be achieved by the concepts of finance andeconomics, which were developed in Marshallian comparative static equilibrium models and notin the currently fashionable dynamic partial di¤erentiation equations, which are, for example,used in stochastic dynamic asset valuation. Despite some di¢culties we encounter with such anunashamed copy - cat approach, the current underdeveloped status of the liquidity analysis of thefinancial markets forces us us to break new grounds so that we can make slow but steady progressin this di¢cult area of risk measurement and analysis of dynamic cash flow markets. The fundamental cash growth rate accounting framework has already been developed and itsempirical applicability has been established by Karnosky and Singer (1994, 1995) and by Los(1998a & b), who extended it to Markowitz’ classical mean - variance portfolio optimization andrisk attribution framework (Los, 2001). It is easy to extend it further beyond mean - varianceanalysis, taking account of the latest developments in stable distribution and extreme value theo-ries. In this paper, we’ll focus on the term structure of so - called laminar and turbulent financialcash flows. By way of several examples we’ll demonstrate the empirical applicability of these newideas and compute, for example, the coe¢cient of illiquidity (= ”kinetic viscosity of cash flows”)for a term structure based on the S&P500 stock market index. This is a direct follow - up of ourearlier, more philosophical remarks about the importance of time - frequency analysis of financialtime series for financial management (Los, 2000a). In another paper, we discuss the theory of financial turbulence, starting with Kolmogorov’s1941 explanation for homogeneous fluid turbulence. In that companion paper, we’ll draw uponexamples from meteorology, as originally suggested in Los (1991) and we’ll describe the Galerkinprocedure, based on the wavelet multiresolution analysis (MRA) to solve the Navier - Stokesequations of financial turbulence, which may become central to the study of turbulence in theglobal financial markets. One important point we should emphasize: financial turbulence is not a ”negative” phe-nomenon, in contrast to financial catastrophes or crises. Turbulence is an e¢ciency enhancing, 4
  6. 6. deterministic, dynamic phenomenon, that we have to understand better for a full appreciation ofthe ongoing proper functioning of the financial markets. Financial turbulence is a limited phe-nomenon that can be observed under particular conditions of well - functioning financial markets,such as in the foreign exchange markets of the Deutschemark, the Euro or the Japanese Yen, whenthese markets are coping with di¤erentials in liquidity. In other words, financial turbulence, which occurs in anti - persistent financial markets, mustbe sharply di¤erentiated from financial crises, which are unpredictable discontinuities and trulycatastrophic crisis phenomena occurring in highly persistent financial markets. Such financialcrises we either need to prevent by changing the institutional structure of the existing financialmarkets to make them less persistent and more liquid, or against which we need to insure andhedge by trading financial catastrophe bonds or derivatives, to prevent their truly catastrophicconsequences, as suggested by Chichilnisky and Heal (1992) and Chichilnisky (1996). Can a financial crises, which emerge in persistent fianancial markets, like real estate, stock,bond and commercial loan markets, generate financial turbulence in interlinked anti - persistentfinancial markets, like the foreign exchange and cash markets? Yes, it can and it does, as we’veobserved in several Asian financial markets in 1997 - 1998.2 Dynamic Financial Cash Flow TheoryWhen financial cash is in motion, its properties are described at each point in time by the propertiesof its cash flow rate or volume flux.Definition 1 The financial cash flow rate, or cash volume flux, ¢X(t), is the product ofthe cash invested (= cash position) X(t ¡ 1) at the beginning of time period t and its cash rate ofreturn (= cash velocity) x(t): ¢X(t) = x(t)X(t ¡ 1) (1)since the value of an asset grows as X(t) = X(t ¡ 1).ex(t) (2) 5
  7. 7. so that (approximately) X(t) ln = x(t) X(t ¡ 1) ¢X(t) = (3) X(t ¡ 1)Example 2 According to the exact cash accounting framework, at time t an investor has threepossible investment instruments: (1) investment in an asset k in country i with rate of returnrik (t), (2) a cash swap with rate of return cj (t) ¡ ci (t), with cj (t) being the risk free cash rate incountry j into which the nominal is swapped, and ci (t) being the risk free cash rate in country i outof which the nominal is swapped and (3) the foreign currency appreciation rate fj (t) of countryj.4 Thus, one particular bilateral investment strategy at time t is represented by the followingstrategic rate of return xijk (t) = rik (t) + [cj (t) ¡ ci (t)] + fj (t) (4)with i, j 2 Z2 being the indices for bilateral cash flows between the various countries and k 2 1Z2 being the index for the various assets (stocks, bonds, real estate, commodities, etc.). Noticethat such an international investment strategy is, according to the Capital Asset Pricing Model,equivalent to the sum of the asset risk premium in local market i [rik (t) ¡ ci (t)] and the cash returnon currency j, [cj (t) + fj (t)] (Los, 1998, 1999; Leow & Los, 1999). This leads to the following definitions of continuous and steady cash flows.Definition 3 Continuous financial cash flow occurs in a particular cash flow channel, when itscash flow rate ¢X(t) is constant ¢X(t) = X(t ¡ 1)x(t) = constant (= independent of time) (5)Remark 4 This equation is also called the cash continuity equation.Definition 5 Steady financial cash flow occurs in a particular investment channel, when its cashflow return rate x(t) = constant (= independent of time). One would expect, under conditions of continuous cash flows, that the cash flow return ratex(t) to be high when the cash flow channel is constricted, i.e., where the investment positionX(t ¡ 1) is small, and the cash flow return rate x(t) to be low where the channel is wide, i.e.,where the investment position X(t ¡ 1) is large.Example 6 Day traders, who have very short investment horizons, trade very quickly with smallinvestment positions, in the order of a $1, 000 and reap highly fluctuating returns x(t) with verylarge amplitudes |x(t)|. In contrast, institutional investors, such as pension funds, insurance funds,or large mutual funds, who usually have long term investment horizons, trade much slower withvery large investment positions in the order of $100 million, and who invest for steady rates ofreturn x(t) with moderate amplitudes |x(t)|. The trillion dollar research question is: can such 4 In such cash accounting frameworks, one usually adopts the US dollar as the base currency, or numéraire. 6
  8. 8. di¤erent types of traders, with investment positions of di¤erent size and investment horizons ofdi¤erent lengths, coexist in the same global markets without financial turbulence, or is that anecessary and unavoidable consequence? Since day traders add liquidity and reduce the persistenceof a financial market, it is likely that they reduce the possible occurrence of financial crises, which isa phenomenon occurring in illiquid, persistent financial markets, when suddenly large investmentpositions are set up or unwound.2.1 Perfectly E¢cient Dynamic Financial MarketsThe motion of empirical cash flows in globally interconnected financial markets is very complex andstill not fully understood, because the flows start and stop at various times and go back and forthbetween millions of bilateral positions. Until very recently, financial economists have proceeded bymaking several heroic, simplifying assumptions for stationary, and often static, markets. Followingthe classical definitions of ideal, perfectly e¢cient physical flows, the following is my suggestionfor a new theoretical definition of perfectly e¢cient dynamic financial markets, which can providea standard for the comparative measurement of dynamic cash flows in non - stationary markets.Definition 7 Perfectly e¢cient dynamic financial markets have cash flows, which are: (1) perfectly liquid (or nonviscous): there is no internal (institutional) friction in themarkets and all assets of di¤erent maturities in all markets can be instantaneously purchased andliquidated; (2) steady: the rate of return x(t) of each cash flow remains constant; (3) incompressible: the density of the cash transactions remains constant in time, i.e., thevolume of market buy and sell transactions remains uniform over time; (4) irrotational: there is no angular momentum of the cash flows, i.e., all cash flows ina particular investment channel at a time t are in the same direction. All decision makers withdi¤erent time horizons for investment receive the same information at the same time and interpretthat information correctly and simultaneously (= the market’s rational ”herd instinct”). Thus, in the ideal situation of a perfectly e¢cient dynamic financial market, the cash flows inthe various investment channels are all streamlined in the same direction (there are no cross - overtrades) and move at the same constant flow rates x(t). There are no ”bulls” (positive investors)and ”bears” (negative investors) trading at the same maturity levels. This means, for example,that the term structure of interest rates can only move parallel to itself and is not allowed to rotate.This is, of course, an abstraction from empirical financial market reality, where rotations in theterm structures of interest rates are a, di¢cult to model, empirical phenomenon (cf. Chapmanand Pearson, 2001). 7
  9. 9. 2.2 Financial Pressure and Cash Flow PowerUnder such abstract, perfectly e¢cient dynamic financial market conditions, one can formulate aBernoulli equation for cash flows which quantifies the concept of financial pressure.5Definition 8 (Bernoulli equation for cash flows) In perfectly e¢cient dynamic financialmarkets 1 F P (t) + ½x2 (t) = constant (6) 2 1where F P (t) is the financial pressure at time t and ½ = $2 is the uniform cash density.Remark 9 The Bernouilli cash flow equation is also called the cash energy equation. This Bernoulli equation for cash flows states that the sum of financial pressure F P (t) and 1 2the kinetic energy of the cash flow per dollar, 2 ½x (t), is constant along a cash flow investmentchannel (streamline). This Bernoulli equation for cash flows describes a Venturi cash flow channelto measure the di¤erence in financial pressure between two di¤erent places along an investmentcash flow channel, as follows. The Bernoulli equation implies that 1 1 F Pi (t) + ½x2 (t) = F Pj (t) + ½x2 (t), where (i, j) 2 Z2 i (7) 2 2 jso that the di¤erence in financial pressure F Pi (t) ¡ F Pj (t), measured simultaneously at twodi¤erent points i and j in a particular investment cash flow channel at time t, is the di¤erencebetween the kinetic energies of the uniform cash flow at these two points: ¢ij F P (t) = F Pi (t) ¡ F Pj (t) 1 £ 2 ¤ = ½ xj (t) ¡ x2 (t) i 2 1 = ¡ ½¢ij x2 (t) (8) 2When the financial pressure at ”upstream” point i is higher than at ”downstream” point j, thecash flow velocity at ”downstream” point j is higher than at ”upstream” point i, and vice versa. 5 Daniel Bernoulli’s most famous work, Hydrodynamics (1738) is both a theoretical and practical study ofequilibrium, pressure and velocity of fluids. He demonstrated that as the velocity of fluid flow increases, itspressure decreases. 8
  10. 10. Financial pressure is relatively high where investment cash flow moves slowly, and is relatively lowwhere it moves fast. This point about the cash flow pressure di¤erential becomes clearer, when we take accountof the size of the investment positions. When the cash flow is continuous, as we thus far haveassumed, thus, when xi (t)Xi (t ¡ 1) = xj (t)Xj (t ¡ 1) = constant (9)so that Xi (t ¡ 1) xj (t) = xi (t) (10) Xj (t ¡ 1)then 1 ¢ij F P (t) = ¡ ½¢ij x2 (t) 2 1 £ 2 ¤ = ½ xj (t) ¡ x2 (t) i 2 " # µ ¶2 1 Xi (t ¡ 1) = ½ xi (t) ¡ x2 (t) i 2 Xj (t ¡ 1) "(µ ¶2 ) # 1 Xi (t ¡ 1) 2 = ½ ¡ 1 xi (t) 2 Xj (t ¡ 1) "( ) # 1 Xi2 (t ¡ 1) ¡ Xj (t ¡ 1) 2 = ½ 2 x2 (t) i (11) 2 Xj (t ¡ 1) 2 2 Note that ¢ij F P (t) > 0 if and only if Xi (t ¡ 1) > Xj (t ¡ 1). This financial market cash flowsystem operates as follows. Consider a particular very crowded asset investment market, withmany market participants, where a lot of cash is squeezed together. As soon as an extra investmentchannel is opened and cash begins to exit that particular asset market, the squeezing, or pressureF P , is least near this small exit, where the motion (= cash flow rate of return x(t)) is greatest.Thus, under the assumption of perfectly e¢cient dynamic financial markets, the lowest financialpressure occurs where the investment channel is smallest and the cash return rate is highest, e.g.,in emerging financial markets, where the marginal productivity and the risk of capital investmentis highest. The highest financial pressure occurs where the investment channel is largest and the 9
  11. 11. cash return rate is lowest, e.g., in developed financial markets, where the marginal productivityand the risk of capital investment is lowest.Remark 10 One should be on the alert for financial turbulence when the di¤erence in financialpressure between the upstream developed and downstream emerging financial markets, ¢ij F P (t), 2is large, because the di¤erence in the cash investments in each of the two connected markets Xi (t¡ ³ ´21)¡Xj (t¡1) is very large, so that the kinetic energy x2 (t) = Xj (t¡1) xi (t) in the underdeveloped 2 j Xi (t¡1)downstream market is very large, since the uniform kinetic energy term 1 ½x2 (t) figures prominently 2in the financial Reynolds number (see below). A high Reynolds number empirically measures theonset of financial turbulence. Multiplying the kinetic energy of the uniform cash flow by its flow rate provides the rate atwhich the energy is transferred, i.e., it provides its power.Definition 11 The power of a cash flow at time t is defined by ¸ 1 2 P ower(t) = ½x (t) [x(t)X(t ¡ 1)] 2 1 = ½x3 (t)X(t ¡ 1) (12) 2 Therefore, the available cash flow power per dollar invested is measured by P ower(t) 1 = ½x3 (t) (13) X(t ¡ 1) 2and the average available cash flow power per dollar invested is 1 © ª ½E x3 (t) (14) 2 © ª Notice that these expressions involve the third moment (skewness) m3 = E x3 (t) of thedistribution of the cash flow rates of return in the term structure of spot rates. For (symmetric)Gaussian investment rates returns m3 = 0. For asymmetric distributions of investment rates ofreturn m3 6= 0. Thus, the average available cash flow power per dollar invested is nil, when thedistribution of the rates of return is symmetric around zero, e.g., when it is as likely to makemoney as it is to lose it. The average cash flow power is positive when the distribution of the ratesof return is positively skewed, and negative when their distribution is negatively skewed. 10
  12. 12. 3 Illiquidity and Financial TurbulenceIn the imperfect empirical international financial markets, in which asset investors with di¤erentcash investment horizons - for example, day traders, bank treasurers and pension fund managers -trade, friction can occur between the various investment cash flows, which show a great diversityof changing strategic cash flow rates of return xijk (t). Such friction can cause either financialturbulence, or financial crises, depending on the degree of persistence of the financial markets asmeasured by the persistence of these time series of the strategic rates of return. Now that we havedefined continuous and steady cash flows, let’s, initially rather informally, define turbulent cashflow.Definition 12 Turbulent financial cash flow is irregular cash flow characterized by small whirlpool- like regions, vortices, ”eddies ” ¢X(t) = x(t)X(t ¡ 1) = irregular, with possible vortices (15)Remark 13 This e¤ectively means that the cash flow rates of return series x(t) is irregular.We’ve already characterized and measured various degrees of irregularity by computing the fractalspectrum of financial cash flow rates based on the Lipschitz-®L and the corresponding fractaldimensions (Karuppiah and Los, 2000). But what can we imagine a cash flow vortex or ”eddy” to be? We suggest the followingdefinition.Definition 14 A financial cash flow vortex occurs, when the fractal di¤erences ¢d x(t) firstrapidly increase in density (become quickly more rapid) and then rapidly decreases in density(become quickly less rapid). In other words, a cash vortex occurs„ when the fractional di¤erenti-ation (in particular the second order di¤erentiation, or convexity) of x(t) is non - homogeneousand there exists a fractal spectrum of irregularities. Thus cash flow vortices are regions of non - homogeneous fractional di¤erentiation of thespot rates of return x(t), which can lead to period - doubling and intermittent behavior. Perioddoubling and intermittency will be discussed in the next chapter. But, it shows up in the recordsof financial time series as clustering of the increments of the Fractional Brownian Motion (FBM).For an example, notice the vortex in the Philippine Pesos after the Thai baht crisis on July 2nd,2001 in Fig. 1. 11
  13. 13. 3.1 Illiquidity: Cash Flow ViscosityWhy would such financial turbulence, consisting of a series of financial vortices, occur? The phys-ical theory of flow dynamics may provide us with some clues, or, at least, with some quantifiableentities, which can measure when turbulence occurs, even though no completely acceptable modelor theory of financial turbulence exists. The term viscosity is used in flow dynamics to characterizethe degree of internal friction, or illiquidity, in a fluid. Equivalently, we can define in finance:Definition 15 financial cash flow viscosity = fianncial cash flow illiquidity = degree of illiquidity of adjacent financial cash flows (16) This internal cash flow friction is associated with the resistance of two adjacent layers of cashflows to move relative to each other, due to Newton’s Second Law for flowsProposition 16 (Newton’s Second Law) Any force applied to a flow results in an equal butopposite reaction, which in turn causes a rate of change of momentum in the flow. Because of the illiquidity of investment cash flows, part of their steady kinetic energy, rep- Rresented by a finite, integrated amount of energy |x(t)|2 dt < 1 is converted to thermal (= R”random”) energy, represented by an infinite integrated amount of energy |x(t)|2 dt ! 1, whichcan cause cash vortices (”eddies”) on the edges of the adjacent investment cash flow channels. Again, analogously to definitions in physical flow dynamics, we can define a coe¢cient of cashflow illiquidity, which would allow us to measure the degree of illiquidity in the financial markets,as follows.Definition 17 The ex post term structure x¿ (t) of any asset at time t is defined by the ex-pression ¸¿ 1 X(t) x¿ (t) = ¡1 X(t ¡ ¿ ) ln X(t) ¡ ln X(t ¡ ¿ ) = (17) ¿ This is easy to see, since X(t) [1 + x¿ (t)]¿ = (18) X(t ¡ ¿ ) 12
  14. 14. so that, by approximation, ¿ ln [1 + x¿ (t)] = ¿ x¿ (t) = ln X(t) ¡ ln X(t ¡ ¿) (19) It is important to understand that the ex post term structure x¿ (t) of an asset is nonlinearlyrelated to the corresponding cash flows X(t) and X(t ¡ ¿ ), defining the asset and its value. Thisbecomes even clearer in the following examples of the fundamental and derivative instruments usedin the financial markets and the way they are accounted for by the double - entry bookkeepingsystem of market values.Example 18 The fundamental security of a simple (Treasury) bond at time t can be decomposedinto a series of zero coupon bonds. A zero coupon bond, or pure discount bond, with maturity ¿and principal payment B¿ (t), has zero coupons, so that its (discounted) present value is B¿ (t) P B0 (t) = (20) [1 + x¿ (t)]¿Thus, we have, ex post, B¿ (t) [1 + x¿ (t)]¿ = P B0 (t) X(t) = (21) X(t ¡ ¿)A ¿ ¡year spot interest rate at time t is the interest rate of a zero bond ¸¿ 1 B¿ (t) x¿ (t) = ¡1 (22) P B0 (t)This term structure of interest rates is the nonlinear relationship between the spot interestrate x¿ (t) and the maturity ¿ for a particular grade of credit quality of obligations (e.g., bills,notes, bonds) (Vasicek, 1977; Cox, Ingersöll and Ross, 1985; Heath, Jarrow and Morton, 1992;Los, 2001, p. 52).Example 19 In Fig. 2 we’ve plotted the term structure of interest rates, as follows. We define B¿ (t) X(t)the following three coordinates: x ´ x¿ (t) the term structure, y ´ P B0 (t) = X(t¡¿ ) the relativevalue of the bond investment (when 0 < y < 1 it is a premium bond; when y = 1 a par bond; andwhen 1 < y a discount bond), and z ´ ¿ , the maturity in years, so that 1 x ¡ y z = ¡1, 0 < y, 0 < z < 30 (23) Submissions to Journals and Awards/Cash Flow Dynamics/Article Files/GNOKAY00.wmf 13
  15. 15. 1 0.8 0.6 x(t) 0.4 0.2 20 0 15 5 10 10 X(t)/X(t-T) 15 T 20 5 25 30 0 Term structure of interest rates x¿ (t)as function of maturity ¿ and gross investment cash flow change X(t)/X(t ¡ ¿)Example 20 For the second fundamental security of a stock we have the following valuation.For (not necessarily constant) dividend payments D¿ (t) and a cost of capital x¿ (t) at time t, thedividend discount model (DDM) of stock valuation for an ongoing concern (= a firm with aninfinite life time) computes the stock’s present value at time t to be equivalent to an infinite seriesof ”zero coupon bonds” with maturities ¿ = 1, 2, ..., 1 1 X D¿ (t) P S0 (t) = (24) ¿=1 [1 + x¿ (t)]¿Thus for each of the dividend payments we have the cash flow relationship D¿ (t) [1 + x¿ (t)]¿ = P S0 (t) X(t) = (25) X(t ¡ ¿)so that the ¿ ¡year spot dividend yield is ¸¿ 1 D¿ (t) x¿ (t) = ¡1 (26) P S0 (t)Example 21 The rational law of pricing by arbitrage implies that all other financial instru-ments, like derivatives, can be derived from these two fundamental asset valuations by simple 14
  16. 16. linear transformations. Call and put options can be synthesized from a linear portfolio of bondsand stocks (cf. Los, 2001, pp. 164 - 166). Forwards and futures can be represented as time-shiftedbonds or stocks (cf. Los, 2001, pp. 225 - 227). Swaps can be represented by a combination of along and a short bond, or, equivalently, by a series of zero coupon bonds, or a series of long andshort forwards (cf. Los, 2001, pp. 239 - 243).Example 22 The double-entry bookkeeping model of accounting shows that the value of any con-cern can be represented as a linear portfolio of fundamental and derivative financial instruments(cf. Los, 2001, p. 24). The Accounting Identity of the balance sheet with current marketvalues is: Assets(t) = Liabilities(t) + Equity(t) or A(t) = L(t) + E(t) or, rewritten, E(t) = A(t) ¡ L(t) (27)This Accounting Identity is the basic model for exact financial modeling, since financial instru-ments can be viewed as combinations of long and short positions of the fundamental securities.Changes in the net equity position ¢¿ E(t) over the accounting period ¿ , which is usually onequarter of a year or a year, are produced by the corresponding changes in the assets and liabilities,as reflected in the income statement6 ¢E¿ (t) = ¢A¿ (t) ¡ ¢L¿ (t) or xE (t)E(t ¿ ¡ ¿) = xA (t)A(t ¿¡ ¿ ) ¡ xL (t)L(t ¡ ¿ ) ¿ or N et Income¿ (t) = Revenues¿ (t) ¡ Expenses¿ (t) (28)on market value basis, where xA (t) is the market valued Rate of Return on Assets (ROA) over the ¿period ¿ , xL (t) is the market valued rate of debt liability expense (= rate of interest on all debt) ¿over the period ¿, and xE (t) is the market valued Rate of Return on Equity (ROE) over the period ¿¿ . Thus the Accounting Identity combines the bundles of discounted cash flows in a linear fashionand these discounted cash flows are nonlinearly related to the rates of return. In physics, shear stresses are the internal friction forces opposing flowing of one part of asubstance past the adjacent parts. In finance, we are concerned about the shear stresses ofinvestment cash flows with di¤erent degrees of illiquidity adjacent in the complete term structure,flowing past each other within and between the various financial term markets and their maturity”buckets.”Example 23 One can form a (not necessarily square) matrix of bilateral cash flow channelsbetween two countries, so that every cash flow investment channel associated with a particularterm rate of return x¿i (t) is adjacent to every other cash flow investment channel, resulting in anet rate of return for both cash channels together x¿i ,¿j (t) = x¿i (t) ¡ x¿j (t) = xij (t): 2 3 x11 (t) x12 (t) ... x1,T2 (t) 6 x (t) x22 (t) ... ... 7 x(t) = 6 21 4 7 5 (29) ... ... ... ... x¿1 ,1 (t) ... ... x¿1 ,¿2 (t) 6 It is easy to view liabilities as ”negative assets.” 15
  17. 17. Each side of the matrix can represent, say, the term structure of assets in a country, whereby onecountry may have more investment channels and di¤erent asset term structures than another (asis empirically the case), so that this matrix is not necessarily symmetric, or square. For example,the U.S.A. has a very fine term structure with T1 = 30 year bonds, while other countries have onlyT2 = 10 year bonds. Using tensor algebra, Los (1998a & b) provides empirical examples of suchbilateral cash rates of return matrices based on the cash rates and stock market return rates in 10Asian countries plus Germany. But how does cash flow shear stress and strain occur and what do these terms mean?Definition 24 Cash flow shear stress is the change in the ex post asset term structure relativeto the original asset prices, ¢x¿ (t)/X(t ¡ ¿ ). It is the quantity that is proportional to the cashflow supply of earnings on an investment made ¿ periods ago, causing deformation of the termstructure of the asset return rates.Definition 25 Cash flow shear strain is the change in an asset price relative to its maturity¢X¿ (t)/¿ . It is the quantity that is proportional to the cash flow demand. When we express the cash strain per unit of time ¢t = 1 we have the cash rate of return(velocity) gradient.Definition 26 The cash rate of return (velocity) gradient is Cash flow shear strain ¢X¿ (t)/¿ = ¢t ¢t x¿ (t) = (30) ¿ This gradient is the cash flow rate of return of an asset with maturity ¿ relative to its term, asa measure of the degree of term structure deformation caused by cash flow shearing, i.e., caused bythe di¤erences between the cash flow rates of return with di¤erent time horizons for investment.This gradient is conceptually similar to the laminar velocity profile of the hydraulic flow in pipes. For small cash flow stresses, stress is proportional to strain and we can define a coe¢cientof cash flow illiquidity, or kinetic cash viscosity, based on this simple linear stress - deformationrelationship.Definition 27 The coe¢cient of cash flow illiquidity or kinetic cash flow viscosity attime t is ¯ ¯ ¯ cash flow shear stress ¯ ¯ ´¿ (t) = ¯ ¯ cash flow shear strain ¯ ¯ ¯ ¯ ¢x¿ (t)/X(t ¡ ¿ ) ¯ =¯ ¯ ¯ x¿ (t)/¿ ¯ ¯ ¯ ¯ ¢x¿ (t)¿ ¯ =¯¯ ¯ (31) x¿ (t)X(t ¡ ¿ ) ¯ 16
  18. 18. This relative illiquidity coe¢cient measures the change in the term structure ¢x¿ (t) multipliedby its term (maturity) ¿ relative to the cash flow rate x¿ (t)X(t ¡ ¿ ). When the investment cashflow is continuous, i.e., when the investment cash flow rate x¿ (t)X(t¡ ¿ ) is constant, all illiquidityis measured by the empirical changes in the term structure ¢x¿ (t) for each term ¿ . When ¢x¿ (t) varies inversely with the term ¿ , the illiquidity coe¢cient is constant, ´¿ (t) =´¿ . Another way of expressing this, is to state ¢x¿ (t)¿ = ´¿ x¿ (t)X(t ¡ ¿) (32)Thus, for laminar cash flow, the ”force of illiquidity resistance” is ¢x¿ (t)¿ _ x¿ (t)X(t ¡ ¿ ) (33)i.e., is proportional to the rate of cash return x¿ (t) for maturity ¿ and the size of the original cashinvestment X(t ¡ ¿ ). Still a di¤erent way of interpreting this relative illiquidity coe¢cient is tostate that it measures the absolute relative change in the interest rates of a particular maturity¢x¿ (t) X(t¡¿ ) x¿ (t) , relative to the size of the average invested cash flow ¿ in particular maturity market: ¯ ¯ ¯ ¢x¿ (t) ¯ ¯ x¿ (t) ¯ ´¿ (t) = ¯ X(t¡¿) ¯ ¯ ¯ (34) ¯ ¿ ¯ After these introductory definitions - which can all be calculated from the basic data - we arein the position to define perfect cash flow liquidity.Definition 28 Perfect cash flow liquidity exists when ´¿ (t) = 0. That occurs either (1) when¢x¿ (t) = 0 for finite invested cash flows X(t ¡ ¿ ) and finite investment horizon terms ¿, implyingthat the finite term rates of return structure remains unchanged under financial stress, thus x¿ (t) =constant; or (2) when, unrealistically, the term rates of return are infinite x¿ (t) = 1 for all finitecash flows and finite terms; or when the cash flow for a finite term ¿ and a finite term structurex¿ (t) is infinite X(t ¡ ¿ ) = 1; or when the term equals zero, ¿ = 0, for a finite investment cashflow X(t ¡ ¿ ) and a finite spot rate of return x¿ (t).Remark 29 Most real world cash flows are non - perfectly liquid with ´¿ (t) > 0, because mostreal world cash flows are term dependent. There is not much instantaneous cash in comparison.Also, infinite term rates of return and invested cash flows empirically don’t exist, nor do zeroinvestment horizons. Thus the only realistic outcome is that perfect cash flow liquidity exists when¢x¿ (t) = 0 for finite invested cash flows X(t ¡ ¿ ) and finite investment horizon terms ¿. 17
  19. 19. This particular expression for the liquidity coe¢cient is only really valid when the term struc-ture x¿ (t) varies linearly with the term ¿ , i.e., when the rate of cash return gradient x¿ (t)/¿is uniformly constant. When this is the case, one may speak of Newtonian cash flows, in whichstress, illiquidity, and rate of strain are linearly related. This is, of course, seldom the case with thedynamic term structures in the international financial markets. Empirically, the term structuresof the cash investment markets vary nonlinearly, in such a way that the short term cash rates ofreturn vary relatively more (= have larger amplitudes), and vary more frequently, than the longterm cash rates of return.7Remark 30 Term structure modelers have attempted to capture this nonlinear dynamics by,first, intercorrelating the various terms x¿ (t) for a limited number of terms 1 < ¿ < T , and,then, by analyzing the resulting covariance matrix using a static principal component, or factoranalysis (cf. the survey by Chapman and Pearson, 2001). The objective of such spectral analysisis to capture most of the variation of the term structure by retaining a small number, say three,principal component factor loadings, like Factor 1 = Level, Factor 2 = slope, and Factor 3 =Curvature of the term structure. But such inexact identification scheme of a covariance matrix isinherently subjective, because which eigenvalues are to be considered significant and to be retainedand which should be considered representing noise? (For a detailed discussion of such ”prejudices”of principal components, cf. Los, 1989). When the rate of return gradient is not uniform, as is usually the case, we must express theilliquidity coe¢cient in the general marginal form: ¯ ¯ ¯ ¢x¿ (t)/X(t ¡ ¿ ) ¯ ´¿ (t) = ¯ ¯ ¯ ¯ @x¿ (t)/@¿ ¯ ¯ ¯ ¢x¿ (t)@¿ ¯ =¯ ¯ ¯ @x¿ (t)X(t ¡ ¿ ) ¯ (35)Thus, we compare the empirical change in the required cash flow over an infinitesimal small changein the term, ¢x¿ (t)@¿ , with the infinitesimal change in the available cash flow rate @x¿ (t)X(t¡¿ ).That is a comparison between the marginal investment cash flow shear stress and the marginalinvestment cash flow shear strain.8 7 Professor Craig Holden of Indiana University has made is a simple historical demonstration of this nonlinearlyvarying term structure: ”Craig Holden’s Excel-based Interactive ”Movie” of Term Structure Dynamics ” (1996),which can be downloaded from his web page. 8 We use the partial derivatives @ to indicate that we’re interested in variation on the term structure x (t) ¿caused by a marginal change in the term ¿ at a fixed time t. 18
  20. 20. 3.2 Laminar and Turbulent Financial Cash FlowsWe will now define the important concepts of laminar and turbulent cash flows.Definition 31 If the adjacent layers of illiquid term cash flow smoothly alongside each other ina financial market, and the term cash flow rates x¿ (t) are constant, the stable streamline flow ofinvestment cash through a particular financial market (of a particular term and risk profile), iscalled laminar.Definition 32 If the streamlined term cash flows in a financial market, at su¢ciently high termrates of return x¿ (t), change from a laminar cash flow to a highly irregular cash flow with highlyirregular cash flow (velocity) rates x¿ (t), it is called turbulent. Both, laminar and turbulent flows are easily illustrated by the hot cigarette smoke, whichinitially is laminar, but then turns turbulent due to the nonlinear constraints imposed by theinteractive heat exchange with the cooler environment. (Fig. 16 in Schroeder, 1991, p.26). Onecan measure such cash flow velocity rate changes by a spectrogram or a scalogram. My current conjecture is that the nonlinear expectation constraints inherent in the logical termstructure x¿ (t) cause such chaos in the cash flows in financial markets to emerge, when particularparametric thresholds in the interlocking term structures are surpassed. In other words, particularshapes of interlocking term structures, at su¢ciently high term rates, can cause irregularities toemerge.Remark 33 Small, perfectly liquid, laminar cash flow motions are called acoustic cash flow mo-tions and (linear) Fourier Transform analysis can be used to decompose such steady and continuousglobal cash flow motions in subcomponent flows. However, when any of these conditions do notapply, and we deal with large - scale, so - called non-acoustical cash flow motions, like cash flowturbulence, we need (nonlinear) Wavelet Transform analysis to analyze and decompose them. At this moment, analogously to physical turbulence measurements, I also conjecture that theonset of cash flow turbulence can be identified by a dimensionless parameter, called the Reynoldsnumber for uniform cash flow. In hydrodynamics and aerodynamics, the Reynolds number is adimensionless ratio related to the velocity at which smooth flow shifts to turbulent flow.9 Thereis no fundamental reason why we should not be able to measure a similar Reynolds numbers forfinancial cash flows. 9 Osborne Reynolds (1842 -1912), was an English scientist whose papers on fluid dynamics and turbulenceremain basic to turbine operation, lubrication, fluid flow and streamlining, cavitation, and the e¤ects of tides. 19
  21. 21. Definition 34 The financial Reynolds number Re¿ (t) for a uniform cash flow of maturity term¿ is given by ¯ ¯ ¯ a½x¿ (t) ¯ Re¿ (t) = ¯ ¯ ¯ ´¿ (t) ¯ ¯ ¯ ¯ ¯ ¯ ½x¿ (t)X(t ¡ ¿) ¯ =¯ ¯ ¢x¿ (t)¿ ¯ ¯ ¯ x¿ (t)X(t¡¿ ) ¯ ¯ 2 ¯ ¯ ½x¿ (t)X 2 (t ¡ ¿) ¯ =¯¯ ¯ (36) ¢x¿ (t)¿ ¯ 1where ½ = $2 is again the uniform cash flow density, which renders the Reynolds number dimen-sionless, and a = X(t ¡ ¿) is the scale of the investment cash flow channel. The beauty of the Reynolds number is that it measures dynamic similarity: flows with thesame Reynolds number look the same, whereas flows with di¤erent Reynolds numbers look quitedi¤erent. It is a measure of the amount of cash flow present at a particular time in a particularchannel. When the cash flow Reynolds number is constant, Re¿ (t) = Re¿ , for turbulent investmentcash flows show ½x2 (t)X 2 (t ¡ ¿ ) ¿ ¢x¿ (t)¿ = (37) Re¿ Thus, for turbulent cash flow, the ”force of illiquidity resistance” is a quadratic term. It isproportional to the squares of the rate of cash return and the size of the cash investment, ¢x¿ (t)¿ _ x2 (t)X 2 (t ¡ ¿ ) ¿ (38)This quadratic expression has a much higher value than the ”force of illiquidity resistance” orviscosity, for laminar cash flow, which we know to be linearly proportional ¢x¿ (t)¿ _ x¿ (t)X(t ¡ ¿ ) (39)Remark 35 The cash flow Reynolds number is proportional to the uniform cash flow energy1 22 ½x¿ (t) at the ( ”downstream ”) measurement point. When the financial pressure ”upstream ” ishigh, this ”downstream ” energy increases rapidly, producing a large Reynolds number. Turbulentcash flows have violent and erratic fluctuations in velocity and pressure, which are not associatedwith any corresponding fluctuations in the exogenous forces driving the flows. Physical turbulence is generally considered to be a manifestation of the nonlinear nature ofthe underlying fundamental equations. The ”force of illiquidity resistance” is a quadratic, there-fore highly nonlinear, term. We will see in Chapter 10, that this quadratic term is the essential 20
  22. 22. constraint ”causing ” deterministic chaos or turbulence to occur at particular values of the growthrate parameters of the underlying simple dynamic processes. Feedback loops of the simple dy-namic investment cash flow processes quickly increases the complexity of the aggregated dynamicinvestment cash flow process of several interconnected financial markets.Remark 36 The physical flows in round pipes are laminar for Re¿ (t) < 2000, but physical tur-bulence begins to occur for Reynolds numbers, Re¿ (t) > 3000. On the other hand, in physicalturbulent flow over flat surfaces Re¿ > 500, 000. We don’t know yet what the critical values arefor the investment cash flow Reynolds numbers. This is an important issue for the empiricalresearch into financial turbulence we’re currently conducting at Kent State University. Since the scale of the investment cash flow channel is measured by a uniform dollar a = 1$, wehave a½ = 1 , so that the Reynolds number for uniform cash flows can be interpreted as a term $based return/liquidity risk ratio: ¯ ¯ ¯ x¿ (t) ¯ Re¿ (t) = ¯ ¯ ¯ ´¿ (t) ¯ (40)The higher the cash rate of return on an investment asset with maturity ¿, and the lower thecash illiquidity (= the higher the cash liquidity) for that same maturity, the higher the Reynoldsnumber and, thus, the opportunity for financial turbulence.10 In other words, the financial Reynolds number measures the cash rate of return on an invest-ment relative to its illiquidity risk, which is a quite di¤erent, and, perhaps, a much more importantmeasure than the usual market volatility risk ¾¿ (t). It’s a matter of current empirical financial research to determine at which magnitudes ofthis cash flow Reynolds number financial turbulence begins to occur and if it is a real empiricalproblem. But this expression shows that the Reynolds number for finite term rates of returnx¿ (t) only becomes very large when the illiquidity risk approaches zero, ´¿ (t) ! 0. Thus financialturbulence is a phenomenon of very liquid financial markets and not of illiquid markets. In fact, 1 0 The value of an asset is computed as the sum of the discounted (future) cash flows associated with that asset.Each asset has a particular maturity, even though it is sometimes assumed to have an infinite maturity, as in thecase of, for example, the assets of a firm, based on the accounting principle of an ”ongoing concern.” However, inreality, any firm has finite assets, i.e., projects with finite maturities. 21
  23. 23. the e¢ciency - enhancing financial turbulence contrasts sharply with the high risk discontinuityphenomena of the very illiquid and e¢cient markets (which show very low Reynolds numbers).Remark 37 Indeed, the current empirical evidence in Chapter 11 shows that turbulence is likelyto occur in the anti - persistent anchor currency markets of the D - Mark and the Yen, relative tothe US dollar.3.3 Financial Wavelet Reynolds Numbers and Financial IntermittencyHow can we translate this financial Reynolds number in terms of the wavelet MultiresolutionAnalysis (MRA) of Chapters 7 and 8? Recall from Chapter 7 the definition of the scalogram ofthe cash rate of investment return x(t). Then, analogously to Farge et al. (1996), we have thefollowing definition:Definition 38 The local financial wavelet spectrum SW (¿, a) of x(t) is defined as its scale -standardized scalogram PW (¿, a) SW (¿, a) = a |W (¿, a)|2 = (41) aRemark 39 With the suggested financial uniform scale a = 1$ that standardization is very simpleand SW (¿, a) = |W (¿, a)|2 /$ (42) A characterization of the local ”activity ” of x(t) is given by its wavelet intermittency, whichmeasures local deviations from the mean spectrum of x(t) at every time lag ¿ and scale a. It isa local measure of financial risk, i.e., a measure of risk localized in the scale (frequency) - timedomain. Thus we no longer need the idealized statistical concept of ergodicity of Chapter 1 tomeasure financial risk. We can locally measure financial risk. While we provide already here away of quantitatively measuring intermittency, the concept of intermittency has been explainedin greater detail in Los (2000b), when we discuss the various phases of degeneration into chaos ofnonlinear financial pricing processes.Definition 40 The financial wavelet intermittency of x(t) is defined as 2 |W (¿, a)| IW (¿, a) = R (43) R |W (¿, a)|2 d¿ 22
  24. 24. Remark 41 One advantage of this wavelet intermittency measure is that it is dimension - free. Next, we implement the suggestion by Farge et al. (1996, p. 651) to express the Reynoldsnumber in terms of wavelets, as follows.Definition 42 The financial wavelet Reynolds number is defined as ¯ ¯ ¯ a½ |W (¿, a)| ¯ ReW (¿, a) = ¯ ¯ ¯ ¯ ´¿ (t) ¯ ¯ ¯ a |W (¿, a)| ¯ ¯ ¯ =¯ 0.5 ¯ (44) ¯ ´¿ (t)Và ¯ 0.5so that ½ = 1/Và , where the constant Z Và = |ÿ,a (t)|2 d¿ da (45) R2is the uniform variance of the analyzing wavelet ÿ,a (t) of the scalogram W (¿, a) over all timehorizons ¿ and scales a.Remark 43 Our expectation is that at large (inverted frequency) scales a, the wavelet financialReynolds number coincides with the usual large - scale Reynolds number Re¿ (t). In the smallestscales (say a = ¾" . where ¾" is the Kolmogorov scale of turbulent cash flow, as in Chapter 11),one expects this wavelet Reynolds number to be close to unity, when averaged over time. Analogously to Farge et al. (1996), the current empirical research question regarding theanalysis of financial turbulence is the following. With such a wavelet Reynolds number defined fortime lag ¿ and scale a, are there time lags ¿ where such a Reynolds number at some very smallscale is much larger than at others, and how do such time periods correlate with time periods ofsmall - scale activity within the cash flow? If so, then ReW (¿, a) could give an unambiguous answerof the activity at small scales (or at any desired scale a). Such time periods of high ReW (¿, a) canthen be interpreted as periods of strong nonlinearity, i.e., periods of financial turbulence. In other words, if the interpretation of Farge et al. (1996) is correct, we can just look at”scalograms of financial Reynolds numbers” to detect periods of strong nonlinearities and thus offinancial turbulence.11 11 At Kent State University, we are currently creating such financial scalograms. 23
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